Geodesic
Distributions and Mnemofunctions on Adeles. Fourier Transform
Informatics and Automation, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 228-240.

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Some classical results are recalled, and a finite part distribution is interpreted as the zero-order term in the expansion of a homogeneous distribution. An adelic finite part distribution and a generalization of the Tate distribution are defined, and their Fourier transforms are calculated. The machinery of mnemofunctions (nonlinear generalized functions) is adapted to $p$-adic and adelic cases, and product formulas for some specific distributions are given. 
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E. M. Radyno; Ya. V. Radyno. Distributions and Mnemofunctions on Adeles. Fourier Transform. Informatics and Automation, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 228-240. http://geodesic.mathdoc.fr/item/TRSPY_2004_245_a22/

[1] Antonevich A. B., Radyno Ya. V., “Ob obschem metode postroeniya algebr obobschennykh funktsii”, DAN SSSR, 318:2 (1991), 267–270 | MR | Zbl

[2] Gelfand I. M., Graev M. I., Pyatetskii-Shapiro I. I., Teoriya predstavlenii i avtomorfnye funktsii, Obobschennye funktsii, Nauka, M., 1966, vyp. 6 | MR

[3] Vladimirov V. S., Volovich I. V., Zelenov E. I., $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR

[4] Volovich I. V., Radyno Ya. V., Khrennikov A. Yu., Obobschennye funktsii i operator Vladimirova na gruppe konechnykh adelei Vybranyya navukovyya pratsy Belaruskaga dzyarzhaunaga uniiversiteta, 6, BDU, Minsk, 2001

[5] Egorov Yu. V., “K teorii obobschennykh funktsii”, UMN, 45:5 (1990), 3–40 | Zbl

[6] Radyno E. M., “Mnemofunktsii egorovskogo tipa na $\mathbb{Q}_p$”, Dokl. NAN RB, 47:4 (2003), 10–13 | MR

[7] Radyno Ya. V., Ngo Fu Tkhan, Sabra R., “Preobrazovanie Fure v algebre novykh obobschennykh funktsii”, Dokl. RAN, 327:1 (1992), 20–24 | Zbl

[8] Colombeau J.-F., New generalized functions and multiplication of distributions, North-Holland, Amsterdam, 1989 | MR

[9] Dragovich B., “On generalized functions in adelic quantum mechanics”, Integral Transforms and Spec. Funct., 6:1/4 (1998), 197–203 | DOI | MR | Zbl

[10] Radyno Ya. V., “Extensions of algebras, mnemofunctions and their applications”, Nonlinear theory of generalized functions, Proc. Workshop (Viena, 1997), Chapman Hall/CRC Res. Notes Math., 401, Chapman Hall, New York etc., 1999, 209–218 | MR | Zbl

[11] Haran S., “Riesz potentials and explicit sums in arithmetics”, Invent. Math., 101 (1990), 697–703 | DOI | MR | Zbl